$a = r\left(\ctg{\frac{\beta}{2}}+\ctg{\frac{\gamma}{2}}\right) = \dfrac{r\cos{\frac{\alpha}{2}}}{\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}}}$

$S=(p-a)^{2} \operatorname{tg} \frac{\alpha}{2} \operatorname{ctg} \frac{\beta}{2} \operatorname{ctg} \frac{\gamma}{2}$

$\sin{\alpha}+\sin{\beta}+\sin{\gamma} = \dfrac{p}{R}$

$\cos{\alpha}+\cos{\beta}+\cos{\gamma} = \dfrac{R+r}{r}$

$\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}} = \dfrac{r}{4R}$